Warm-up:  p 185 #1 – 7

Section 12-3: Infinite Sequences and Series In this section we will answer…  What makes a sequence infinite?  How can something infinite have a limit?  Is it possible to find the sum of an infinite series?

WW hat kind of sequence is it? FF ind the 18 th term. NN ow find the 20 th, 25 th, and 50 th. SS o …the larger n is the more the sequence approaches what? Consider the following sequence: 16, 8, 4, ….

Sum of an Infinite Geometric Series  In certain sequences, as n increases, the terms of the sequence will decrease, and ultimately approach zero.  This occurs when ______________.  What will happen to the Sum of the Series?

Sum of an Infinite Geometric Series The sum, S n, of an infinite geometric series for which is given by the following formula:

Example #1  Find the sum of the series:

Example #2  A tennis ball dropped from a height of 24 feet bounces 50% of the height from which it fell on each bounce. What is the vertical distance it travels before coming to rest?

Example #3  Write … as a fraction using an Infinite Geometric Series.

Try another…  Write …as a fraction using a geometric series.

Limits  Limits are used to determine how a function, sequence or series will behave as the independent variable approaches a certain value, often infinity.

Limits  They are written in the form below:  It is read “The limit of 1 over n as n approaches infinity”.

Limits  They are written in the form below:  It is read “The limit of 1 over n as n approaches infinity”.  To evaluate the limit substitute infinity for n:

Possible Answers to Infinite Limits  You may get zero or any number.

Possible Answers to Infinite Limits  You may get infinity.  That means no limit exists because it does not approach any single value.  You may get no limit exists because the sequence fluctuates.

Possible Answers to Infinite Limits  You may get infinity over infinity.  This is indeterminate; meaning in its present form you can’t tell if it has a limit or not.

Possible Answers to Infinite Limits  You may get infinity over infinity.  This is indeterminate; meaning in its present form you can’t tell if it has a limit or not. Let’s do some test values…

Possible Answers to Infinite Limits  You may get infinity over infinity.  This is indeterminate meaning in its present form you can’t tell if it has a limit or not. Let’s do some test values… This approaches 1/3 but how do I prove it?

Algebraic Manipulation of Limits  Method 1: Works only if denominator is a single term. – 1) If denominator is single term, split the into separate fractions. – 2) Reduce – 3) Take Limit

Algebraic Manipulation of Limits  Method 2: This works for all infinite limits. – 1) Divide each part of the fraction by the highest power of n shown. – 2) Reduce. – 3) Take limit (Some terms will drop out).

Limits  Use the fact that to evaluate the following:

The Recap:  What makes a sequence infinite?  How can something infinite have a limit?  Is it possible to find the sum of an infinite series?

Homework:  P 781 # 15 – 39 odd, 40